After digging through the bowels of digikey, I have picked the microchip MCP3561 for the analog to digital converter. It is a 24 ADC with a maximum sampling rate of 153.6kHz and can talk over SPI at 2MHz. It can also perform cross cycle redundancy checks to make sure that the data was not mangled in the transmission.
It would be nice to have a slightly faster sampling rate, but a comparable ADC at 250kHz is almost 10x more expensive. For a 60uS current spike we will still get almost 10 samples.
The datasheet suggests that I supply the reference voltage using an MCP1501 high precision buffered voltage reference, which sounds like a good idea.
For the microcontroller, I think it is important for it to be arduino compatible so that the device is community friendly. Since it needs to be fast and is going to interface with a computer, the teensy 3.2 seems like a good option. This will also distribute the cost of the device - people can use the teensy for other things if they want to.